8.1 Introduction to differentiation8.1 Introduction to differentiation Open the podcast that accompanies this leaflet Introduction This leaflet provides a rough and ready introduction

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8.1

Introduction to differentiationOpen the podcast that accompanies this leaflet

IntroductionThis leaflet provides a rough and ready introduction to differentiation. This is a techniqueused to calculate the gradient, or slope, of a graph at different points.

1. The gradient functionGiven a function, for example, y = x2, it is possible to derive a formula for the gradient of itsgraph. We can think of this formula as the gradient function, precisely because it tells us thegradient of the graph. For example,

when y = x2 the gradient function is 2x

So, the gradient of the graph of y = x2 at any point is twice the x value there. To understandhow this formula is actually found you would need to refer to a textbook on calculus. Theimportant point is that using this formula we can calculate the gradient of y = x2 at differentpoints on the graph. For example,

when x = 3, the gradient is 2 × 3 = 6.

when x = −2, the gradient is 2 × (−2) = −4.

How do we interpret these numbers ? A gradient of 6 means that values of y are increasing atthe rate of 6 units for every 1 unit increase in x. A gradient of −4 means that values of y aredecreasing at a rate of 4 units for every 1 unit increase in x.

Note that when x = 0, the gradient is 2 × 0 = 0.

Below is a graph of the function y = x2. Study the graph and you will note that when x = 3 thegraph has a positive gradient. When x = −2 the graph has a negative gradient. When x = 0the gradient of the graph is zero. Note how these properties of the graph can be predicted fromknowledge of the gradient function, 2x.

When x = 0 the gradient is zero.x

y

5

10

15

When x = -2 the gradient is negative

and equal to -4.

When x = 3 the gradient is positive

and equal to 6.

-4 -3 -2 -1 0 1 2 3 4

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Example

When y = x3, its gradient function is 3x2. Calculate the gradient of the graph of y = x3 whena) x = 2, b) x = −1, c) x = 0.

Solution

a) when x = 2 the gradient function is 3(2)2 = 12.

b) when x = −1 the gradient function is 3(−1)2 = 3.

c) when x = 0 the gradient function is 3(0)2 = 0.

2. Notation for the gradient functionYou will need to use a notation for the gradient function which is in widespread use.

If y is a function of x, that is y = f(x), we write its gradient function asdy

dx.

dy

dx, pronounced ‘dee y by dee x’, is not a fraction even though it might look like one! This

notation can be confusing. Think ofdy

dxas the ‘symbol’ for the gradient function of y = f(x).

The process of findingdy

dxis called differentiation with respect to x.

Example

For any value of n, the gradient function of xn is nxn−1. We write:

if y = xn, thendy

dx= nxn−1

You have seen specific cases of this result earlier on. For example, if y = x3,dy

dx= 3x2.

3. More notation and terminology

When y = f(x) alternative ways of writing the gradient function,dy

dx, are y′, pronounced ‘y

dash’, ordf

dx, or f ′, pronounced ‘f dash’. In practice you do not need to remember the formulas

for the gradient functions of all the common functions. Engineers usually refer to a table knownas a Table of Derivatives. A derivative is another name for a gradient function. Such a tableis available on leaflet 8.2. The derivative is also known as the rate of change of a function.

Exercises

1. Given that when y = x2, dydx

= 2x, find the gradient of y = x2 when x = 7.

2. Given that when y = xn, dydx

= nxn−1, find the gradient of y = x4 when a) x = 2, b) x = −1.

3. Find the rate of change of y = x3 when a) x = −2, b) x = 6.

4. Given that when y = 7x2 + 5x, dydx

= 14x + 5, find the gradient of y = 7x2 + 5x when x = 2.

Answers

1. 14. 2. a) 32, b) −4. 3. a) 12, b) 108. 4. 33.

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